In mathematics, statistics, and computer science, particularly in the fields of machine learning and inverse problems, regularization is a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting. The Tikhonov regularization method is widely used to solve complex problems in engineering. The vertical derivative of gravity can highlight the local anomalies and separate the horizontal superimposed abnormal bodies. The higher the order of the vertical derivative is, the stronger the resolution is. However, it is generally considered that the conversion of the high-order vertical derivative is unstable. In this paper, based on Tikhonov regularization for solving the high-order vertical derivatives of gravity field and combining with the iterative method for successive approximation, the Tikhonov regularization method for calculating the vertical high-order derivative in gravity field is proposed. The recurrence formula of Tikhonov regularization iterative method is obtained. Through the analysis of the filtering characteristics of this method, the high-order vertical derivative of gravity field calculated by this method is stable. Model tests and practical data processing also show that the method is of important theoretical significance and practical value.

中文翻译：

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用Tikhonov正则化迭代方法计算重力场中的高阶垂直导数

在数学，统计学和计算机科学中，尤其是在机器学习和逆问题领域，正则化是引入其他信息以解决不适定问题或防止过度拟合的过程。Tikhonov正则化方法被广泛用于解决工程中的复杂问题。重力的垂直导数可以突出局部异常，并将水平叠加的异常体分开。垂直导数的阶数越高，分辨率越强。然而，通常认为高阶垂直导数的转换是不稳定的。本文基于Tikhonov正则化求解引力场的高阶垂直导数，并结合迭代法进行逐次逼近，提出了计算重力场垂直高阶导数的Tikhonov正则化方法。得到了Tikhonov正则化迭代方法的递推公式。通过对该方法滤波特性的分析，该方法计算得到的重力场的高阶垂直导数是稳定的。模型测试和实际数据处理也表明该方法具有重要的理论意义和实用价值。